Spatial Bayesian Inference

A conditional autoregressive (CAR) prior is a Gaussian Markov random field that makes each area's latent effect conditionally dependent on its neighbors' effects. Use it for areal (polygon) data — administrative regions, census tracts, grid cells — where adjacency defines neighborhood structure. For point-referenced data with exact coordinates, a Gaussian process with a distance-decay covariance function is more appropriate.

INLA (via R-INLA) is faster and often sufficient for latent Gaussian spatial models, making it the default for routine disease mapping or geostatistical interpolation. Full MCMC (e.g., via Stan) is preferable when the model falls outside the latent Gaussian class or when exact posterior samples are needed. Always verify that INLA's Gaussian approximation is adequate by comparing with MCMC on a small example.

Include fixed effects for observed covariates in the linear predictor alongside spatially structured random effects. The random effects capture residual spatial autocorrelation unexplained by the covariates. After fitting, check that Moran's I on the posterior mean residuals is near zero; persistent autocorrelation signals model misspecification.

An exceedance probability map shows, for each location, the posterior probability that its quantity of interest exceeds a policy-relevant threshold. Because Bayesian inference produces a full posterior for each location, these probabilities are a natural output and are more decision-relevant than point-estimate maps alone.

Frequentist spatial regression (spatial lag or error models) provides point estimates with asymptotic standard errors. Spatial Bayesian inference produces full posterior distributions, enables principled information-borrowing through hierarchical priors, handles small-area estimation naturally, and allows complex extensions (space-time, multivariate) in a unified probabilistic framework.